Optimal. Leaf size=8 \[ -2 \text {csch}\left (\sqrt {x}\right ) \]
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Rubi [A] time = 0.19, antiderivative size = 8, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6715, 2606, 8} \[ -2 \text {csch}\left (\sqrt {x}\right ) \]
Antiderivative was successfully verified.
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Rule 8
Rule 2606
Rule 6715
Rubi steps
\begin {align*} \int \frac {\coth \left (\sqrt {x}\right ) \text {csch}\left (\sqrt {x}\right )}{\sqrt {x}} \, dx &=2 \operatorname {Subst}\left (\int \coth (x) \text {csch}(x) \, dx,x,\sqrt {x}\right )\\ &=-\left (2 i \operatorname {Subst}\left (\int 1 \, dx,x,-i \text {csch}\left (\sqrt {x}\right )\right )\right )\\ &=-2 \text {csch}\left (\sqrt {x}\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 8, normalized size = 1.00 \[ -2 \text {csch}\left (\sqrt {x}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.47, size = 37, normalized size = 4.62 \[ -\frac {4 \, {\left (\cosh \left (\sqrt {x}\right ) + \sinh \left (\sqrt {x}\right )\right )}}{\cosh \left (\sqrt {x}\right )^{2} + 2 \, \cosh \left (\sqrt {x}\right ) \sinh \left (\sqrt {x}\right ) + \sinh \left (\sqrt {x}\right )^{2} - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.13, size = 17, normalized size = 2.12 \[ \frac {4}{e^{\left (-\sqrt {x}\right )} - e^{\sqrt {x}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.16, size = 7, normalized size = 0.88 \[ -2 \,\mathrm {csch}\left (\sqrt {x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.31, size = 17, normalized size = 2.12 \[ \frac {4}{e^{\left (-\sqrt {x}\right )} - e^{\sqrt {x}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.75, size = 8, normalized size = 1.00 \[ -\frac {2}{\mathrm {sinh}\left (\sqrt {x}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\coth {\left (\sqrt {x} \right )} \operatorname {csch}{\left (\sqrt {x} \right )}}{\sqrt {x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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